Critical points of multivariables

[duki]

Homework Statement

Find and classify all critical points of
$$f(x,y) = x^3+2y^3-3x^2-3y^2-12y$$

Homework Equations

The Attempt at a Solution

So I've gotten to
$$Fx = 3x^2 - 6x = 0$$
$$Fy = 6y^2 - 6y - 12 = 0$$

$$x=0, x=2, y=-1, y=2$$

Now I can't remember how to find the critical points from here.
Any help is appreciated!

 

[Dick]

The critical points are where Fx=0 AND Fy=0. Why is that hard to remember? Offhand, I would say you have four of them.

 

[duki]

Oh yeah, thanks. And it's hard to remember because I'm still a student.

 

[Mark44]

It's not that much different from how it is with functions of one variable. If y = f(x), you look for values for which f'(x) = 0. With functions of two or more variables, you look for points for which all the partial derivatives are zero.